we mean by a periodic function. The function $f (x)$ is periodic, with the period $a$ , if (i) for any value of $x$ we have $f (x) = f (x + a)$ , and (ii) there is no number $b$ smaller than $a$ such that for any value of $x$ , $f (x) = f (x + b)$ .

The second clause is put into the definition because when we have $\sin \frac{2 π x}{a}$ , it is not only periodic in the period $a$ , but also in the periods $2 a$ and $3 a$ , and so on; this arises since $\sin \frac{2 π (x + 3 a)}{a} = \sin (\frac{2 π x}{a} + 6 π) = \sin \frac{2 π x}{a} .$ So it is the smallest period which we want to get hold of and call the period of the function. The greater part of the abstract theory of periodic functions and the whole of the applications of the theory to Physical Science are dominated by an important theorem called Fourier's Theorem; namely that, if $f (x)$ be a

periodic function with the period $a$ and if $f (x)$ also satisfies certain conditions, which practically are always presupposed in functions suggested by natural phenomena, then $f (x)$ can be written as the sum of a set of terms in the form191

$c_{0} + c_{1} \sin (\frac{2 π x}{a} + e_{1}) + c_{2} \sin (\frac{4 π x}{a} + e_{2}) + c_{3} \sin (\frac{6 π x}{a} + e_{3}) + etc.$

In this formula $c_{0}$ , $c_{1}$ , $c_{2}$ , $c_{3}$ , etc., and also $e_{1}$ , $e_{2}$ , $e_{3}$ , etc., are constants, chosen so as to suit the particular function. Again we have to ask, How many terms have to be chosen? And here a new difficulty arises: for we can prove that, though in some particular cases a definite number will do, yet in general all we can do is to approximate as closely as we like to the value of the function by taking more and more terms. This process of gradual approximation brings us to the consideration of the theory of infinite series, an essential part of mathematical theory which we will consider in the next chapter.

The above method of expressing a periodic

function as a sum of sines is called the "harmonic analysis" of the function. For example, at any point on the sea coast the tides rise and fall periodically. Thus at a point near the Straits of Dover there will be two daily tides due to the rotation of the earth. The daily rise and fall of the tides are complicated by the fact that there are two tidal waves, one coming up the English Channel, and the other which has swept round the North of Scotland, and has then come southward down the North Sea. Again some high tides are higher than others: this is due to the fact that the Sun has also a tide-generating influence as well as the Moon. In this way monthly and other periods are introduced.

We leave out of account the exceptional influence of winds which cannot be foreseen. The general problem of the harmonic analysis of the tides is to find sets of terms like those in the expression on [page]191 above, such that each set will give with approximate accuracy the contribution of the tide-generating influences of one "period" to the height of the tide at any instant. The argument $x$ will therefore be the time reckoned from any convenient commencement.

Again, the motion of vibration of a violin string is submitted to a similar harmonic analysis, and so are the vibrations of the ether and the air, corresponding respectively to waves of light and waves of sound. We are here in the presence of one of the fundamental processes of mathematical physics–-namely, nothing less than its general method of dealing with the great natural fact of Periodicity.